- #1

Vandmelon

- 3

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## Homework Statement

I'm looking at the 1d harmonic oscillator

\begin{equation}

V(x)=\frac{1}{2}kx^2

\end{equation}

with eigenstates n and the time dependent perturbation

\begin{equation}

H'(t)=qx^3\frac{(\tau^2}{t^2+\tau^2}

\end{equation}

For t=-∞ the oscillator is in the groundstate n=0

I need to show that for n>3 the states will not get excited and I only need to look at the first order perturbation.

## The Attempt at a Solution

What I'm thinking is \begin{equation}P_{a→b}=\vert c_b(t)\vert^2 \end{equation} therefore I need to find when c_b=0

and because

\begin{equation}

c_b'=-\frac{i}{\hbar}∫H_{ba}'(t')e^{i\omega_0t'}dt'

\end{equation}

I need to find

\begin{equation}

H_{ba}'=<\psi_n\vert H' \vert \psi_a >=0

\end{equation}

For n>3

It would make sense if H_ba look a bit like this

\begin{equation}

H_{ba}'=\frac{d^n}{dx^n}H'

\end{equation}

But I get some horrible results if I try to solve H_ba. Is it right what I am doing or is it way off.